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Contributions to the arithmetic of imaginary quadratic fields. (Beiträge zur Arithmetik imaginärquadratischer Zahlkörper.) (German) Zbl 0564.10025
Mathematisch-Naturwissenschaftliche Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn (1984); Bonn. Math. Schr. 161, 159 S. (1985).
The purpose of this work is to investigate the connection discovered by F. Grunewald et al. between automorphic forms with respect to \(\mathrm{SL}_2(R)\) and \(\mathrm{GL}_2(R)\), where \(R\) is the ring of integers of an imaginary quadratic field \(K\), and \(Q\) the set of elliptic curves over \(K\) with good reduction everywhere. In the first part the author develops an ingenious method of Harder to obtain bounds for the ’cuspidal’ part, \(H^1_{\text{cusp}}(\mathrm{SL}_2(R),Q)\), of \(H^1_2(\mathrm{SL}_2(R),Q)\). He determines when this group is trivial and also gives upper and lower bounds for the dimension of this space. The arguments in this part are geometrical and use the Lefschetz fixed point theorem. He also computes \(\dim H^1_{\text{cusp}}((\mathrm{SL}_2(R),Q)\) for those \(R\) whose discriminant does not exceed 100.
In the second part the author determines those imaginary quadratic fields over which there is an elliptic curve with good reduction almost everywhere ‘almost completely’ and also determines this set ‘almost completely’. The proofs consist in reducing the problem to a diophantine one (following ideas of Setzer and Stroeker) and in analyzing this problem carefully.

MSC:
11F12 Automorphic forms, one variable
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R39 Langlands-Weil conjectures, nonabelian class field theory
14H52 Elliptic curves
11R11 Quadratic extensions